\chapter[Numbers and Sets \\ \textnormal{\emph{Lectured in Michaelmas \oldstylenums{2020} by \textsc{Prof.\ I.\ B.\ Leader}}}]{Numbers and Sets}
\emph{\Large Lectured in Michaelmas \oldstylenums{2020} by \textsc{Prof.\ I.\ B.\ Leader}}

This course gives an introduction to university level maths.
We begin by presenting some basic rules, called axioms.
Then, we carefully prove logical statements that follow from these axioms.
We build upon our previous results iteratively until we have proven some important theorems.
Almost every other course in an undergraduate maths course will follow this pattern, and this course acts as a prototypical example.

In the first part of the course, we rigorously define the natural numbers, integers, rationals, and reals.
Using the axioms, we can prove facts about things like prime numbers, modular arithmetic, and limits.
In the second half of the course, we establish the notion of a set, and define what concepts like functions are.
At the end, we use the rules of sets to prove that there are different sizes of infinity.

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